The chemical difference between RNA and DNA is the missing 2'-hydroxyl group in the nucleotides that build DNA. The major effect of that change that I know of is the higher stability of DNA compared to RNA. But I'm wondering whether this difference has significant implications for the ability of DNA to form compex, three-dimensional structures.

RNA is known to be able to from complex tertiary structures and function as ribozymes. It clearly has the ability to form a wide range of structures and can catalyze a variety of chemical reactions.

As far as I know, there are no naturally occuring catalytic DNAs known. But a number of synthetic DNA enzymes have been created in the lab, so it is generally possible for DNA to form catalytic structures (see Breaker and Joyce 1994 for the first created DNA enzyme).

I'm wondering whether the missing 2'-OH means that DNA has less potential to form complex structures compared to RNA? I imagine it changes the ability to create hydrogen bonds, but I don't know if it would significantly decrease the potential structures that DNA could adopt.

3 Answers
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To make sure I'm not comparing apples and pears, my (attempt to) answer the question will be broken into two parts: comparison of single-stranded nucleic acids and double stranded ones.

Single stranded DNA and RNA

Both DNA and RNA can form single-stranded complex tertiary structures in which the secondary structure elements are associated through van der Waals contacts and hydrogen bonds. The presence of a 2'-hydroxyl group makes ribose ring prefer different conformations than deoxyribose in DNA. Also, since 2′-OH moiety is both a hydrogen donor and acceptor, it provides RNA with greater flexibility to form 3D complex structures and stability to remain in one of these conformations. As Aleadam notices, this paper shows that tRNA and its DNA analog form similar tertiary structures though tDNA is not as stable as tRNA:

Therefore, we submit that the global conformation of nucleic acids is primarily dictated by the interaction of purine and pyrimidine bases with atoms and functional groups common to both RNA and DNA. In this view the 2-hydroxyl group, in tRNA at least, is an auxiliary structural feature whose role is limited to fostering local interactions, which increase the stability of a given conformation.

These authors also show that at least one loop in the tDNA analog is more susceptible to cleavage by a restriction endonuclease. In this region the tRNA has a water molecule hydrogen bonded to 2'hydroxyl group.

I was not able to find more of such interesting comparisons in the literature.

Double stranded DNA and RNA

Both DNA and RNA can form double-stranded structures. Again, sugar conformation determines the shape of the helix: for DNA helix it's usually B-form, whereas helical RNA forms A-geometry under nearly all conditions. In RNA helix we find the ribose predominantly in the C3’- endo conformation, as 2'-OH stericly disfavors the C2'-endo conformaion, necessary for B-form geometry.

Physiological significance

dsRNA and ssDNA often provide a signal to the cell that something is wrong. dsRNA is of course seen in normal processes like RNA interference but it can also stop protein synthesis and signal viral infections (cf. double stranded RNA viruses). Similarly, ssDNA is much more prone to degradation than dsDNA, it often signals damage of DNA, or infections from single stranded DNA viruses and induces cell death. Therefore, due to their functions, under normal conditions DNA 3D structure is mostly a double-stranded helix, whereas RNA has a single stranded, "protein-like", complex 3D structure.

This answer is incorrect for a variety of reasons. For one, it makes the assessment that RNA is more flexible. It isn't; DNA is. The role described in the Aleadam paper thus is minimal. The comments about degradation has little to do with the ability to adopt 3D structures. It is more of a reflection of the presence of specific defensive mechanisms via DNases and RNases.
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bobthejoeMar 3 '14 at 21:01

This is not my field so I'm risking a wrong/incomplete answer here, but I'd say that the critical difference is the almost complete occurrence of double-stranded DNA that precludes the formation of the tertiary structures in single-stranded RNA, rather than the 2'OH difference. In fact, and following the link you posted, the authors even comment in the introduction that:

"It is well known that single-stranded DNA can assume interesting tertiary structures. A tRNA and its DNA analog form very similar structures [9]".

The answer lies entirely in the thermodynamic stability that is provided with having a 2'-OH. As mentioned by Aleksandra, RNA will adopt only the C3'-endo conformation whereas DNA adopts both the C2'-endo and C3'-endo. Effectively, this makes the DNA strand more flexible not RNA. In doing so, a single stranded DNA oligomer will be able to adopt more states.

DNA/RNA helix formation is dominantly enthalpically driven. When a helix forms, RNA will only adopt an A-form Helix where as DNA will adopt both a A-form and B-form. While there are more possible conformations for DNA, the reduction in the entropic contributions make it significantly more unfavorable. Interestingly, this is why RNA analogs like PNA and morpholinos have good binding properties as they will form more entropically stable base-pairing with their target sequence.

For these reasons, it is much more common so see structured Ribozymes and non-coding RNAs in nature even though it is physically possible to produce DNAzymes. Again, one of the many reasons why the RNA world hypothesis makes sense.